Evolution, prime numbers, and an algorithm for the creative process
1 2 3
Shi Huang 4
5 6
Center for Medical Genetics & Hunan Key Laboratory of Medical Genetics, School of Life 7
Sciences, Central South University, Changsha, Hunan, China 410078 8
huangshi@sklmg.edu.cn 9
10 11 12
Key words: evolution, Darwinism, molecular clock, neutral theory, MGD theory, prime 13
numbers, Riemann Hypothesis, uniqueness, uniformity, creativity, the Prime Law 14
15
Abstract 16
Recent understandings of molecular evolution, together with the fossil records, have 17
established that there are both linear and nonlinear processes in the creation of novel species, 18
which is strikingly similar to the generation of prime numbers and human creativity. Each 19
creation of a more complex species is like a prime number, unpredictable, discontinuous, and 20
yet can be modeled by a smooth curve in relation to time. The mystery behind the complexity 21
increases in nature and human civilizations might well turn out to be similar to that behind the 22
appearances of prime numbers. Here we show that an algorithm for the creative process of 23
humans can create prime numbers in a lawful and yet unpredictable fashion. The essence of 24
primes is the duality of uniqueness and uniformity together with the creation algorithm. The 25
algorithm consists of the non-linear process of uniformity selection to create the unique and 26
the linear process of uniqueness selection to form the uniformity. The iterations of this 27
algorithm can create an infinite number of primes. The algorithm appears to have been 28
hardwired in the human brain as shown by recent experimental studies. This new 29
understanding can deduce some of the best-known properties of primes and may explain the 30
nearly constant and yet seemingly random creation of novelty in relation to time.
31 32 33 34 35 36 37 38
Introduction:
39
Evolution and evolutionary theories 40
To use the concept of evolution to explain the seemingly endless creations of species 41
in life history was largely made popular by the works of Darwin and Wallace. The concept 42
was inspired by observing the phenotypic adaptations of living species as well as the fossil 43
records. Phenotypic variations were thought to be randomly generated, which are followed by 44
natural selection to either keep the fit or eliminate the unfit variants. While many have argued 45
that natural selection is not a random process, no one has disputed that the appearance of an 46
environmental condition, such as a hot weather, is anything but random.A single random 47
event is enough to make the outcome of a chain of events random. The selector in the natural 48
selection processes is always random in a Darwinian intention-less world. The natural 49
selection process from the selector hot weather to the elimination of heat sensitive variants is 50
surely non-random. But the selector hot weather is random, which makes the end results of 51
natural selection random. Calling the process non-random is not wrong but is meaningless.
52
The end result is what matters, which can only be either random or intentional. The natural 53
selection process is non-random but the end result could still be random if the selector is 54
randomly caused. Thus, the creation of novelty as explained by Darwin and Wallace is 55
strictly a linear process of creation by chances or accidents.
56
The chance creation theory of Darwin and Wallace, while does explain well 57
microevolution or small scale changes such as the appearance of drug resistance in bacteria, 58
has long found difficulty, or has not met with any evidence, in explaining macroevolution or 59
large scale changes in complexity such as the formation of the first life from inorganic 60
materials or the advance from single cell organisms to multicellular species. Furthermore, the 61
theory has been challenged by the most astonishing phenomenon in evolution at the level of 62
genomes or molecules as first discovered in the early 1960s, the genetic equidistance 63
phenomenon [1-3]. For any three or more species of different organismal complexity as 64
intuitively defined by the number of cell types, one can perform two kinds of sequence 65
alignment. The first aligns a complex organism such as human against simpler or less 66
complex species that evolved earlier such as frogs and fishes. The second aligns simpler 67
organisms such as fishes against those more complex ones such as chickens and humans. The 68
first kind of alignment shows the pattern that human shares more identity with chickens than 69
with fishes or a hierarchy of increasingly less identity to increasingly less complex species, 70
which is largely consistent with Darwinian expectations [1]. Margoliash in 1963 performed 71
both alignments and made a formal statement of the molecular clock hypothesis after noticing 72
the genetic equidistance result where fishes are equidistant to chickens and humans or simple 73
species are equidistant to all more complex species, which is unexpected from Darwin‟s 74
theory [2, 4]. In hindsight, however, Margoliash has mistakenly converted a maximum 75
saturation phenomenon that can vary across species and populations into a linear-phase 76
phenomenon.
77
The nearly constant and similar mutation rate (i.e., molecular clock) interpretation of 78
the equidistance phenomenon has in fact turned out to be a classic tautology since it has not 79
been verified by any independent observation and has on the contrary been contradicted by a 80
large number of facts [5-10]. Nonetheless, researchers had initially treated the molecular 81
clock as a genuine reality and had in turn proposed a number of theories to explain it [11-16].
82
The „Neutral Theory‟ has become the favorite [14-16], even though it is widely 83
acknowledged to be an incomplete explanation for the clock [9, 17]. The observed rate is 84
measured in years but the Neutral theory predicts a constant rate per generation. Also, the 85
theory predicts that the clock will be a Poisson process, with equal mean and variance of 86
mutation rate. Experimental data have shown that the variance is typically larger than the 87
mean. Ohta‟s “nearly neutral theory” explained to some extent the generation time issue by 88
observing that large populations have faster generation times and faster mutation rates but 89
remains unable to account for the great variance issue [18]. With the neutral and nearly 90
neutral theory, molecular evolution has been treated as the same as population genetics or 91
microevolution. However, the field still lacks a complete theory as many have acknowledged 92
[19, 20], and has unfortunately yet to pay attention to the equidistance result, which has been 93
considered by some as “one of the most astonishing findings of modern science”[21, 22].
94
We recently proposed the maximum genetic diversity (MGD) hypothesis to explain 95
the genetic equidistance phenomenon based on a pair of intuitions or axioms [8, 23-25].
96
Axiom 1 posits that the more complex the phenotype, the greater the restriction on the 97
choices or errors in molecular building parts. Axiom 2 says that any system can allow a 98
limited level of random errors or noises in molecular building parts and such errors may be 99
beneficial, deleterious, or neutral depending on circumstances. Obviously, one only needs to 100
substitute “errors in molecular building parts” for “genetic diversity” to get the equivalent 101
concepts in biology. Axiom 2 in effect underlies the proven virtues of the modern evolution 102
theory consisting of Darwin‟s and Kimura‟s theories. It is because species have built-in 103
robustness or allowed range of random errors in the first place that chance events could lead 104
to limited order. If an organism is built in a way that cannot allow any random errors in its 105
genome, chance would be only destructive and Darwinian mechanisms would not be able to 106
work at all to create order.
107
Genetic diversity or distance cannot increase indefinitely with time and has a 108
maximum limit being restricted by function and physiology or epigenetic complexity. The 109
MGD of simple organisms is greater than that of complex organisms. Over long evolutionary 110
time, the genetic distance between sister species and a simpler outgroup (more distant) taxon 111
is mainly determined by the MGD of the simpler outgroup, although over short time scales it 112
is determined by time, drift, environmental selection, and the neutral mutation rates of the 113
simpler outgroup as well as to a smaller extent by the rates of the sister taxa. The MGD 114
hypothesis thus includes the proven virtues of modern evolution theory, consisting of 115
Darwin‟s theory and the neutral theory, as relevant only to microevolution over short time 116
scales before sequence divergence reaches MGD. An increase in epigenetic complexity 117
during macroevolution is associated with a suppression of genetic diversity or point mutations.
118
So, evolution involves two distinct processes, linear or microevolution versus nonlinear or 119
macroevolution (Fig. 1). The linear process is largely just accumulating random mutations 120
followed by drift or natural selection. The nonlinear process is a sudden and dramatic change 121
or increase in the complexity of epigenetic programs that necessarily demands a reduction in 122
MGD or the maximum tolerable level of random errors in the genomes. The timing for the 123
epigenetic change is not arbitrarily or randomly set but is determined by the time required for 124
the prior species to reach MGD or maximum mutation saturations. The origin of life from 125
inorganic materials can be likened as a reduction in entropy as life building molecules must 126
lose degree of freedom when turning from its inorganic existence into existence in a life form.
127
According to the MGD theory, such entropy reduction underlies not only the origin of the 128
first life but also the origin of each and every major advance in complexity in the 129
macroevolution of species.
130
The MGD hypothesis explains the genetic equidistance phenomenon as a result of 131
maximum genetic distance imposed by physiological or epigenetic constraints [8, 23-25].
132
This phenomenon has in fact another characteristic, the overlap feature where particular sites 133
in an amino acid sequence are subject to multiple different mutational changes in a particular 134
lineage which has been overlooked for nearly half of a century [26]. While the molecular 135
clock may superficially explain the apparent equidistance in quantities, it cannot explain the 136
non-random distribution of mutation hot spots and the related observation that the percentage 137
of constrained sites in more complex clades is greater than that in simpler organisms. The 138
MGD theory has accounted for major phenomenology of molecular evolution. It has also 139
been instrumental in directing productive research into not only evolutionary phylogenetic 140
problems but also key biomedical problems [26-37].
141
In a sequence alignment with humans, there is a hierarchy with humans less and less 142
related to increasingly less complex species (Fig. 2). As less complex species evolved earlier, 143
the hierarchy of gene identities shows correlations with two different parameters, complexity 144
and time. If one only focused on the time correlation, one would conclude that protein 145
non-identity is only determined by time of separation as if the substitution rate is constant and 146
the same among species (hence the molecular clock). On the other hand, if one focused on the 147
complexity parameter and ignored time, one would find a strong correlation of sequence 148
identity with species complexity. One also finds that simple species is equidistant to all more 149
complex species. So, the distance hierarchy with humans as measured by fast evolving 150
proteins at maximum saturation distance is a result of lower and lower complexity of species 151
in more ancient times and hence increasingly higher within-species MGD. The saturation 152
distance to human for a lower complexity species is equal to the within-species MGD of the 153
lower taxon.
154
Genomes have two types of sequence mismatches, functional and neutral, both of 155
which show correlation with time. The neutral variations are explained by the neutral theory 156
and show correlation with time during the linear phase of evolution. The functional variations 157
are correlated with physiology, as explained by the MGD theory, and indirectly with time as 158
the degrees of physiological complexity are correlated with time with simple physiology 159
evolved earlier in time.
160
The molecular clock or constant rate interpretation is really about the nearly constant 161
rate of complexity increases. People since Aristotle have long appreciated the direction of 162
evolution towards higher complexity. Scholars believing in Darwin‟s theory have always 163
denied this but only by ignoring or misreading inconvenient facts such as the genetic 164
equidistance phenomenon. The evidence for complexity increases is commonplace and easy 165
to notice. The first molecular evidence for it is the maximum genetic equidistance 166
phenomenon. What is most striking is the nearly constant rate of complexity increases as 167
measured in years, which can be quantitatively shown by the fraction of non-changeable 168
positions in a protein or the fraction of identical residues between human and a lower 169
complexity species (Fig. 2). Such nearly constant rate is inconsistent with chance creations as 170
chance would mean much greater irregularity: events may cluster at some time windows 171
while rarely happen at some other time windows. The challenge is how to prove that it is not 172
chance.
173 174
Prime numbers 175
As nature is written in the language of mathematics, it would be most unusual if a 176
fundamental natural phenomenon, i.e., the nearly constant rate of evolution towards higher 177
complexity as measured in years (Fig. 2), has no counterpart in mathematics and vice versa.
178
An intriguing analogy is the pattern of prime numbers [38]. The cumulative increase in prime 179
numbers along the progression in natural numbers is well known to follow a smooth curve 180
with a nearly constant rate, especially when viewed from a distance (Fig. 3) [39, 40]. Here the 181
progression in natural numbers is like a time clock, rigid and predictable. The appearance of 182
prime numbers is discontinuous like a staircase and unpredictable but follows nonetheless a 183
well-defined function Li(N) as shown by the Riemann hypothesis [38]. Such a pattern is 184
inconsistent with chance as chance would mean much greater irregularity. Each new 185
appearance of a more complex species is like a new prime number, unpredictable, 186
discontinuous, and yet nearly constant. Individual species are well known to appear in the 187
fossil record abruptly as evidence for the punctuated equilibrium model of macroevolution 188
has shown [41]. However, the discontinuous appearance of species of higher and higher 189
complexity still follows a very smooth and regular pattern as shown by the equidistance 190
phenomenon (Fig. 2). The striking similarity between evolution and prime numbers suggests 191
that there may be a common explanation underlying both.
192
The German mathematician Bernhard Riemann formulated the Riemann Hypothesis 193
(RH) in 1859 [42]. The hypothesis is widely regarded as the most important unsolved 194
problem in all of mathematics. The RH is believed by most mathematicians to be true. A 195
large number of deep and important other results have been proven under the condition that it 196
holds. The RH essentially says that the primes are as regularly distributed as possible given 197
their seemingly unpredictable occurrence on the number line. According to the Prime Number 198
Theorem of Gauss, the number of primes less than N is approximately the logarithmic 199
integral Li(N) or less precisely N/ln(N). If the RH is true, the error between Li(N) and the true 200
number of primes is at most of the order of the square root of N [38, 40, 43-45]. This error 201
margin is the smallest possible and cannot be improved by much [46]. This is the error 202
margin expected by the theory of probability for some unpredictable events such as a coin 203
toss. Random means no pattern and yet mathematics is largely about finding and proving 204
patterns. The difficulty with the RH probably lies in the fact that it requires one to prove the 205
absence of a pattern, which has rarely if ever been done in mathematics. How does one 206
demonstrate unpredictable or “absence of a pattern” as wondered by Tao [47]? Is it even 207
possible to do?
208
A prime number is commonly defined as a positive integer that has only two divisors, 209
1 and itself. Both the number 1 and 2 can be either included or excluded as primes by 210
manipulating the definition of primes. Accordingly, the primality of the number 1 and 2 are 211
decided by human agreements rather than objective logic or reason. The number 1 is not 212
considered a prime today but was in the past [48-52]. While 2 is considered a prime today, at 213
one time it was not [53]. The odd primes have many properties not shared by 2, the only even 214
prime. It is also easy to have a definition based on calculation that would include all primes 215
except 2. Thus, a prime can be defined as a positive integer that cannot be expressed by the 216
even number of sums of any single number except 1 and itself. For example, 1 is 0 (an even 217
number) sum of 1 and itself; 3 is 2 sums of 1 and 0 sum of itself; but 2 is not a prime since it 218
is 1 (an odd number) sum of 1.
219
To define numbers by calculation that is itself defined by numbers is a tautology, 220
which merely describes ways of identifying some primes but reveals little about what a prime 221
really is or the essence of primes. A tautological or circular definition necessarily means a 222
lack of true understanding. This leads to the dilemma that 2 is a prime in one definition based 223
on division but not a prime in another equally plausible definition based on addition. It is 224
arbitrary human convenience or taste to favor one tautological definition over another. We 225
can only resolve such dilemma with objective reasoning when we achieve a deeper 226
understanding of primes that is based on knowledge more fundamental than calculation and 227
numbers. Primes are the foundations of mathematics and should have a form of existence or 228
definition that is independent of mathematics.
229
To avoid circularity, a creature must be defined by things that are more basic than it 230
rather than more advanced. We must use quantum particles rather than molecules to define 231
atoms, even though we discovered molecules before we knew about quanta. Just because 232
calculation was discovered before prime numbers in human history does not mean that primes 233
must be defined by calculation. A concept can only be defined by concepts lower or more 234
basic in logical hierarchy. What is even more basic than numbers must be used to define 235
primes and non-primes. If primes are atoms that build other numbers, then the primes must be 236
built by its own building blocks, which would be equivalent to quantum particles.
237
The essence of a creature is its building blocks together with a rule of manipulating 238
the building blocks. The essence is what is ultimately responsible for the properties of a 239
creature. The essence of matter, the quanta building block together with a law of 240
manipulating the quanta, is what is ultimately responsible for the properties of the physical 241
universe. A creature must be defined by its essence. Numbers are creatures of the mind. Are 242
primes related to the creativity of the mind?
243 244
The creative process of the mind 245
The creativity of the human mind is the most remarkable feature of humans that sets 246
humans apart from all other biological species. Comparing today‟s civilization with those of a 247
few thousand years ago, it is clear that humans have been constantly creating things, both 248
physical and metaphysical. While the history of human civilization has seen a countless 249
number of human creations, with the recent creations generally more complex than earlier 250
ones, it seems that the basic capacities of the human brain have remained relatively 251
unchanged at least within the last 5000 years. It seems to be a real phenomenon that more 252
complex things get created over time while the basic capacities of the brain have stayed 253
largely the same. The brain appears to have the ability to know or absorb whatever that have 254
been created before and to come up with novel inventions.
255
There is likely a general pattern or rule that can describe the creative phenomenon of 256
the mind. While creations can be countless and different, the general rule or algorithm 257
employed by the mind for each creation may be the same. Creations by successive 258
generations may be viewed as the iterative applications of the same creation algorithm. To 259
discover that algorithm may be important in order to understand the structures and functions 260
of the brain that seem to be able to absorb past creations and to come up with something new.
261
It may also help to reveal why the mind always becomes bored with new things after a while 262
and why it has an insatiable appetite for novelty.
263
It has not escaped attention by scholars that evolution of species and creative 264
evolution in human civilizations share similarities. The generation of creative ideas is 265
generally viewed as an evolutionary process. Some think it is Darwinian [54, 55], while 266
others not [56]. The question addressed by those studies is how a creative idea evolves from a 267
population of competing ideas within a mind and how insights from evolution of species may 268
help model human creations. Few if any, however, has attempted to independently come up 269
with an algorithm or understanding of the creative evolutionary process of the mind and see 270
how well it may also describe the evolutionary history of species. The question that concerns 271
us here is at a fundamental level: what motivates a mind to create in the first place. The 272
creative process that we would like to analyze is akin to the paradigm evolution process of 273
Kuhn [57]. A paradigm is often initiated by a creative individual and gets established 274
subsequently by countless individuals who make incremental advances within the paradigm.
275
The former is a nonlinear process while the latter linear. For an established paradigm nearing 276
maximum saturation in terms addressing details, some begin to see some major problems 277
while most do not. Then a creative individual comes up with a revolutionary solution to a 278
major problem of the established paradigm and a new paradigm is initiated. The creative 279
process from paradigm to paradigm continues seemingly without end. Here, we aim to 280
develop a general hypothesis of the creative process underlying human creative activities.
281
The creation algorithm that programs the mind to be creative is obviously the 282
foundation of all human creations. Since numbers (positive integers) may represent the most 283
fundamental creatures of the mind, the creation algorithm of the mind should be able to create 284
numbers. We here found that prime numbers can model the creative process of the mind.
285
Others have independently noticed the connection between prime numbers and creation [58].
286
Louis Kauffman and Hector Sabelli observed: “The generation of primes epitomizes the 287
causal creation of novelty." [59]. Don Zagier noted: “Upon looking at these numbers, one has 288
the feeling of being in the presence of one of the inexplicable secrets of creation.” [60].
289
If creation by the mind is lawful or deterministic, i.e., determined by an algorithm, it 290
could be predictable. However, if creation is predictable, it would no longer by definition be 291
novel or unique. For creation to be meaningful to humans, it must not be predictable. Human 292
creativity is also logical and reasoned and does not seem to be arbitrary. Thus, whatever the 293
algorithm that has programmed the human mind and made the mind creative must make the 294
creative process lawful and yet the outcome unpredictable. Similarly, prime numbers have 295
been found to be both lawful and seemingly unpredictable. But a deterministic law of primes 296
or of human creativity remains to be discovered that nonetheless cannot predict the outcomes.
297
Primes are both lawful and seemingly unpredictable. This is highly similar to human 298
creations and appearances of increasingly complex species in macroevolution. Here, we first 299
describe a creation algorithm that makes the creative process lawful but the outcome 300
unpredictable. We then show that this algorithm can create primes in a fully deterministic and 301
lawful process but still allows the primes to have the intrinsic property of unpredictability.
302
The implications of this algorithm to RH, creativity, and evolution in nature are then 303
discussed.
304 305
Results and Discussion:
306
The algorithm that makes the mind creative 307
The most fundamental capacities of a human mind may be to know and to imagine, 308
which are essential to creativity. To know is to recognize the unique from a background of 309
contrasting uniformity and vice versa. To imagine is to think of novel things that do not exist 310
previously. By observing how the human mind creates, we have found an algorithm that 311
programs the mind and makes the mind creative. This algorithm consists of a pair of opposite 312
but complimentary yin and yang principles with one underlying a linear process and the other 313
a nonlinear process, and a mind that coordinates the interplay of the two principles. A 314
creation or creature is defined as the unique that does not exist previously, is distinguished 315
from all other imagined things, and can exist subsequent to its creation by being able to 316
initiate a population of followers that share a uniform pattern resembling the unique. A 317
creation has the bipolar duality of uniqueness and uniformity. A follower of a creation is 318
defined as the new thing that does not exist previously but shares some uniform property with 319
a prior creation. A creation is a large advance in paradigm while a follower of a creation 320
represents a small step progress within a paradigm.
321
The imagination of a mind is either following the existing patterns of past creations 322
or is based on a novel pattern. How does a novel pattern come to the mind remains a mystery 323
and is of no concern here. A new but meaningless thing or pattern is not a creation because it 324
cannot be uniquely distinguished from other imagined entities, cannot be logically linked with 325
existing patterns, and cannot initiate a following. A great piece of music or book or art 326
initiates a following by existing in the minds of people who are familiar with the piece. A 327
book that was soon forgotten forever is not a creation but is merely a follower of an existing 328
pattern. Existing pattern consists of both past creations and of a default order-less state. The 329
order-less state is the background and driving force for order and pattern. A new thing that is 330
not following any existing ordered pattern but is not uniquely distinguished from the 331
order-less state is still viewed as a follower because it is following the existing order-less 332
pattern. Things that constitute the order-less state include all that cannot be logically linked to 333
any ordered pattern and cannot be uniquely distinguished from others or are equally unique as 334
others.
335
The yang principle for the nonlinear process is uniformity selection that allows the 336
mind to recognize the unique or the creation. Uniformity abolishes individuality and selects 337
for the unique. Uniformity selection drives the creation of the unique. The yin principle for 338
the linear process is uniqueness selection that allows the unique to initiate a population of 339
followers sharing a uniformity pattern resembling the unique. The mind uses this principle to 340
allow the unique to exist or survive subsequent to its creation. Uniqueness selection results in 341
the formation of an ordered uniformity consisting of individuals that are fittest or most 342
adapted to the unique. The process from the unique to a specific uniformity of a population of 343
followers is essential for the unique to exist subsequent to its creation, which further serves to 344
drive the creation of the next unique. The creative process of the mind is the iterative use of 345
the same creation algorithm and an endless cycling process from uniformity to unique to 346
new-uniformity. When the mind sees the unique, the mind strives to fit and follow. When the 347
mind sees uniformity, the mind strives to be unique. All human minds are a unity of different 348
degrees of the yin and yang principles. The nonlinear process requires more work from the 349
mind than the linear process.
350
To create, the mind needs to know what is known previously, which is termed the 351
existing-uniformity. Selection by existing-uniformity allows the mind to know whether 352
something is new with a meaningfully ordered pattern. In addition, all creations begin from 353
the imagination of the mind. Within the imagined world, there exists a unique entity that is 354
distinguished from the imagined-uniformity shared by other imagined entities. To create by 355
uniformity selection is to bring into existence an imagined entity that is distinct from both the 356
existing-uniformity and the imagined-uniformity.
357
The formation of the order-less existing uniformity is by the default of reproduction 358
and the inherent nature of the mind. The mind treats anything that cannot be rationalized with 359
past creations as part of an order-less uniformity. Random brushes on a canvas would belong 360
to the order-less uniformity. The formation of the ordered existing-uniformity requires the 361
principle of uniqueness selection. This selection process selects individuals to follow the 362
unique creation of the past. The followers of a unique creation are essential to the 363
popularization of the unique and the long-term existence of the unique in the form of 364
existing-uniformity. The followers also contribute new variations or incremental advances 365
around the main theme/paradigm of the unique creation, which would form a new level of 366
existing-uniformity essential for triggering the next unique creation. However, the 367
incremental progress made by the followers cannot directly in itself lead to the next unique 368
creation. Creation of the unique represents a discontinuous nonlinear change in paradigm and 369
is fundamentally different from the linear formation of followers.
370
Existing uniformity thus consists of order-less and ordered. Based on the existing 371
uniformity, the mind is able to know whether something is imagined or not yet existing.
372
Among the things imagined, a uniform property may be shared by all except the unique. The 373
unique is the one that has the closest relationship to the existing uniformity but does not 374
belong to any of the existing paradigms. The creation of the unique cannot come as a logical 375
extension of an existing pattern but is nevertheless logically related to existing patterns after 376
the fact of creation.
377 378
Creating primes by the creation algorithm 379
Like creations of the mind, the odd primes including the number 1 also have the dual 380
property of uniqueness and uniformity. A thing is unique if it is not an inherent part of 381
something else and is different from uniformity. A number is an inherent part of a smaller 382
number either because it is needed for the smaller number to have meaning or because it can 383
be expressed as a pattern of a single smaller number >1. The number 2 lacks uniqueness 384
because it is an inherent part of creating the number 1, as evidenced by the existence of 385
civilizations that had invented only 1 and 2 and by the absence of civilizations that invented 386
only 1 but not 2. We need 2 to invent 1 or for 1 to have any meaning. We need both 1 and 2 in 387
order to invent the concept of number. However, we do not need 3 to invent 1 and 2 as there 388
are human groups that had invented 1 and 2 but not 3 [61].
389
All numbers are inherent in the number 1 as patterns of 1s but the property of 390
uniqueness of the odd primes is not inherent in the pattern of 1s. Uniqueness is in contrast to 391
uniformity and cannot exist independent of uniformity. While a prime can be expressed as a 392
pattern of 1s, its uniqueness cannot. Every number (positive integer) can be uniquely defined 393
by a pattern of 1s but this makes every number equally unique. Thus none is unique. The 394
uniqueness of a number is based on the existence of numbers greater than 1 and the existence 395
of non-unique numbers. Primes and non-primes are like odd and even or yang and yin and 396
cannot exist independent of each other. The number 1 is unique since oneness is synonymous 397
with uniqueness. If 1 is unique, then 2 must be non-unique because it is an inherent part of 398
creating the number 1.
399
The opposite of uniqueness is uniformity or not being able to be singled out. A prime 400
also exists in a pattern, e.g., 18 is a pattern of the prime number 3. In such a pattern, the 401
number 3 could not stand out as a unique individual. The uniformity property of a prime 402
makes it possible for other subsequent primes to be uncovered as the unique. The number 23 403
is a prime because it is not a pattern of any other numbers greater than 1. The number 2 is 404
essential for the number 1 to be unique and for other odd primes to be unique. For uniqueness 405
to exist, the uniformity background must co-exist. Two is the first number of non-uniqueness 406
and therefore has some uniqueness property and the related uniformity property. It is the most 407
unique (the first number of non-uniqueness) and the most uniform (present in more patterns 408
of 1s than any other non-unique numbers) among non-unique numbers.
409
If the building block of non-primes is the prime, it is only fair and logical to go down 410
the hierarchy to ask what may be the building block of primes. The building block cannot be 411
a number since prime number is the lowest level a number (positive integer) can be. If 1 is a 412
prime, its building block must be 1 itself. The number 1 is also the building block of all other 413
primes. A prime is a positive integer that can be built in only one way from its building block 414
1 by way of even number of sums of 1 but not of any other numbers greater than 1. How does 415
a creative mind perceive the number 1? Of course, 1 represents uniqueness or oneness or a 416
single smallest point of the whole. One is also uniformity or the single wholeness and is 417
present everywhere or in every number or in every part of the whole. So, 1 embodies the 418
ultimate duality of uniqueness and uniformity. To a creative mind, 1 and the duality are 419
synonymous. The number equivalent of the duality concept is 1. Since 1 is the sole building 420
block of primes, we can also say that the duality is the building block of primes. To use the 421
duality as the building blocks of primes expresses the meaning of 1 as building blocks in a 422
more fundamental way that is directly linked to the creation algorithm. The following shows 423
that the creation algorithm can use the duality as building blocks to create primes. The 424
mathematical model of the creation algorithm is the orderly creation of primes.
425
Postulate 1. The imagined domain All things created by the mind comes from 426
imagination and the imagined world of the mind is termed the imagined domain. The content 427
of this domain consists of an infinite number of the basic building block of numbers, 1. There 428
are infinite number of patterns of 1, each differ by its count of 1s. Each pattern, except that of 429
a single 1, has the uniform property of having a count of 1s that is between two other patterns.
430
The pattern of 2 is between the pattern of 1 and the pattern of 3. Since the contents of the 431
imagined domain has no numbers smaller than 1, the pattern of a single 1 is not in between 432
two other patterns and is therefore unique.
433
Postulate 2. The reality domain The reality domain is where the materialized 434
creations of the mind exist. A prime is generated in the reality domain because of its 435
uniqueness at the time of its creation. It subsequently exists in the reality domain because of 436
its ability to initiate a pattern/uniformity. A prime is defined as a lawful creature of the mind 437
that has the duality of uniqueness and uniformity. A non-prime is defined as a follower of a 438
prime. The mind creates primes by following the two principles of the creation algorithm as 439
postulated above: 1) to generate uniqueness by uniformity selection and 2) to maintain 440
subsequent existence of the unique by uniqueness selection to form uniformity. Uniqueness 441
selection is the process of species formation or forming follower numbers that share 442
properties with the unique. For example, the follower numbers of 3 are 6, 9, 12, . . .3N, which 443
share the uniform property of 3-ness and form the species of 3. A pattern of 1s or number 444
moves from the imagined domain into the reality domain because it is either uniquely 445
recognized by the mind or is necessary to maintain existence of the unique in the reality 446
domain.
447
Creating primes. Prior to the creation of any numbers in the reality domain, the 448
unique number in the imagined domain is 1. So the first goal is to generate 1 as the unique or 449
prime in the reality domain. Since a prime must form a species or pattern in order to exist 450
following its creation, the species of 1 is formed with 1 followed by the next closest number 2.
451
In addition, to express uniqueness requires the simultaneous presence of uniformity. So, the 452
species of 2 is formed to represent uniformity with 2 followed by the next closest number that 453
shares the property of 2-ness, 4. Two is selected to represent uniformity because it is the only 454
other pattern besides 1 that is available in the reality domain at this point when the species of 455
1 has not yet progressed beyond 2. Fig. 4A shows the contents of the reality domain at its 456
time of creation. The prime/uniqueness/1/odd/yang and non-prime/uniformity/2/even/yin are 457
generated simultaneously and cannot exist independent of each other.
458
After the beginning stage of generating the reality domain, the mind is aware of both 459
the imagined domain and the reality domain. By comparing the two domains, the mind is 460
looking for the next prime or unique pattern among patterns in the imagined domain that have 461
no match in the reality domain. This pattern is now 3 and it is unique because it is the 462
smallest while all other patterns share the uniform property of having counts of 1s that are 463
between two patterns. To express 3 as a prime, the species of 3 (3, 6, 9) is formed in the 464
reality domain. To apply the new concept of 3-ness, all species are extended to the 3rd 465
position. The reality domain has now advanced from the beginning stage of 1 and 2 to the 466
next stage of 3-ness (Fig. 4B). At this stage, a number larger than 3 such as 4 expresses only 467
the concept already established such as 2-ness, 2 units of 2-ness. As soon as the stage of 468
3-ness has reached maximum completion, the mind is again ready to look for the next unique 469
pattern remaining in the imagined domain, which is now 5. From the concepts of 3-ness and 470
5-ness, the 4-ness of 4 is now recognized as the intermediate between 3 and 5. By applying 471
the concept of 5-ness and 4-ness, all number species are extended to the 5th position. The 472
species of 5 (5, 10, 15, 20, 25) is formed to express 5 as a prime. As soon as the concept of 5 473
has been applied to maximum completion, the mind is then ready to look for the next unique 474
pattern that remains in the imagined domain, which is 7 (Fig. 4C). In this way of iteratively 475
applying the same creation algorithm, an infinite number of primes can be generated. It is 476
easy to prove this. As reality is made of known primes and their composite numbers, one can 477
always find the next prime by merely finding the unique or smallest number remaining in the 478
imagined domain.
479
As the timing of creating each prime is determined by the time required to reach 480
maximum application of the concept of the previous prime, it is not randomly or arbitrarily 481
determined and hence would necessarily produce a pattern.
482
Because this creation algorithm of the mind can create primes, it is hereafter termed 483
the Prime Law. Since primes have the same property and meaning as creations of the mind, 484
the word „prime‟ and the word „creation‟ are interchangeable or synonymous. Therefore, the 485
„Prime‟ Law also literally means the „Creation‟ Law.
486 487
Creativity and the Prime Law 488
The uniformity selection principle suggests that the mind is capable of converting all 489
that exist into a background upon which to base new imaginations. The uniqueness selection 490
principle suggests that the mind is also programmed to adapt to existing paradigms. A human 491
mind feels the need to fit in with the conventions of society but also feels the need to be 492
unique or different from all other people. As Arthur Schopenhauer put it: “There is in the 493
world only the choice between loneliness and vulgarity.” Humans display polar opposite sides 494
of creation-related character traits that are selected to coexist by the yang and yin principles.
495
The uniformity selection principle values individualism, ambition, adventurism, 496
self-centeredness, and distaste for routine labor, while the opposites are valued by the 497
uniqueness selection principle. Both are essential for creation to go on, and all humans 498
display unity of different degrees of both. Remarkably, experiments have shown that higher 499
levels of right temporal alpha brainwaves enable people to come up with ideas which are 500
further away from the obvious or well-known uses [62]. Such findings may also explain why 501
right-brained or arts people are well known to be more creative [63]. Thus the Prime Law 502
actually has neural basis and appears to be hardwired in the brain. The insatiable appetite for 503
novelty of the human mind may come from it being hardwired with the uniformity selection 504
principle, which thus may account for the constant creation of novelty in human civilizations.
505
The uniqueness selection principle also accounts for the inherent drive of humans to 506
publicize their creative work once they have created something. If they do not work hard to 507
present their creative work to the public and to have their work accepted and followed by 508
others, their work would not count as a complete creation and would have no impact on the 509
creative evolutionary process of humanity. The Prime Law suggests that this drive to have 510
others to accept and follow one‟s own creative work may be essential to human creativity.
511
Future experimental studies should test if this drive may also have neural basis.
512
The novel concept of uniformity selection seems obvious and explains the mind‟s 513
insatiable appetite for novelty. The idea was inspired by death selection in nature or the 514
inherent drive of life to stay away from death. Death is not disappearance of matter but is 515
merely a return of matter from a unique high complexity ordered state (called life) to a 516
uniformity state of less-ordered matter (called death). All life becomes the same in death in 517
terms of matter. Death is order-less uniformity state of life-building molecules. Each human 518
mind has an inherent need to know what is the self or what is special or unique about the self.
519
For the mind to stay away from uniformity/death, the mind needs to be unique. The only way 520
to be unique is to be creative. But in order to be unique and creative, the mind needs to know 521
or learn first what is the uniformity.
522
Uniformity selection describes the large step creations and uniqueness selection 523
describes the small step progresses. Small step progresses are creations within a paradigm.
524
Large step creations are changes in paradigm. It is widely noted that small step progresses 525
cannot add up to large step creations [64]. The Darwinian theory is a creation law by a 526
mind-less process. Mind is not needed in such a law for creation to occur. It is therefore hard 527
to imagine that a mindless creation law can accurately describe the creation process of the 528
mind. Nonetheless, it has been viewed as a Darwinian process of blind-variation and selective 529
retention [54, 55]. While the notion of blind generation of ideas is debatable [56], this 530
Darwinian view is actually a good description of the process of incremental advances within a 531
paradigm.
532
To create (uniqueness) is more than simply making something new. To be new is 533
necessary but not sufficient for a creation. A creation is not only new but is also unique 534
among all potential new things that can be imagined by the mind and its uniqueness lies in the 535
fact that it has the closest relationship to, or is the most smoothly adapted to, the whole 536
pattern of what exist previously. This concept can be illustrated by the prime generation 537
scheme as shown in Fig. 4C, which represents a time of existence that has no concept beyond 538
5-ness. Many numbers are missing in Fig. 4C and can qualify as new, such as 7, 11, 13, 14, 539
16, 17, 18, 19, 21, 22, 23, 24, and all numbers larger than 25. But only 7 is unique because it 540
is the smallest missing number or because all numbers smaller than the unique number belong 541
to what exist previously. A prime number or a creation is the one that is best adapted to the 542
whole pattern of all that exist, whereas a non-prime number or something that is merely new 543
is best adapted to a particular sub-pattern of the whole.
544
Why creation has the property of uniqueness? Because if it were not, the mind would 545
not be able to recognize it from an infinity of choices. The mind does not create or select 546
ideas by throwing a dice. If that were the case, the creations would not have the property of 547
uniqueness, and the mind would be unconscious of the properties of such creatures. Such 548
creatures would lack any coherent logical relationship among them and would not be able to 549
form the uniformity pattern to drive the next creation.
550 551
Indivisibility of primes 552
Uniqueness means that a number is not an inherent part of a smaller number greater 553
than 1. A prime is not a pattern of any smaller number greater than 1, which means indivisible 554
by any smaller number greater than 1. Indivisibility is therefore a secondary property of 555
primes as the unique and should not in and of itself confer primality. The number 2 is 556
indivisible but is not a prime because it lacks the uniqueness essence. It is an inherent part of 557
creating the number 1 as the unique or prime.
558 559
The duality of unpredictability and regularity of primes 560
It is well known that primes seem to exhibit the duality of unpredictability and 561
regularity. Such seemingly impossible unity of extreme opposites is what makes primes so 562
interesting and mysterious [60].However, the first fact of unpredictability remains unproven.
563
It is the seeming randomness or unpredictability that makes the regularity of primes so 564
striking and interesting. There exist a variety of formulas for either producing the Nth prime 565
as a function of N or taking on only prime values. However, all such formulas require either 566
extremely accurate knowledge of some unknown constant, or effectively require knowledge 567
of the primes ahead of time in order to use the formula [65]. They do not really count as 568
prediction. A true predictive formula should not make use of the knowledge of existing 569
primes in order to predict the next future prime.
570
The newly discovered essence of primes can deduce the duality of unpredictability 571
and regularity. When something can be predicted, it must belong to a pattern. As such, it is 572
not unique and hence, by definition, not a prime. The essence of uniqueness rules out 573
prediction of primes as a viable possibility. There is also another easy way to prove this. To 574
predict primes means to predict uniqueness and in turn uniformity since uniqueness needs 575
uniformity to have meaning. Uniformity is made of existing primes. So to predict primes is to 576
predict existing primes, which is a logical non-sense. The uniformity essence of primes 577
demands that the formation of uniformity from a newly created prime or uniqueness must be 578
regular and predictable. So the uniformity forming property of primes gives rise to the 579
regularity of primes. Primes exist as regularly as possible in the uniformity. The creation of 580
primes is fully determined by the orderly formation of uniformity by existing primes. The 581
lawful rather than lawless way of creating primes explains why primes should follow some 582
regularity patterns, such as the Prime Number Theorem. The unpredictability of individual 583
primes explains why such a pattern cannot be completely precise or free of error margins.
584 585
Seeming randomness and real randomness 586
We use the phrase „seeming randomness or deterministic randomness‟ to describe an 587
outcome of a lawful process that is nonetheless unpredictable, like the creation of primes by 588
the Prime Law. A population of such seemingly random outcomes should show a regularity 589
pattern reflecting the lawfulness and regularity in the process leading to these outcomes.
590
However, even the most precise pattern should still show some error margin reflecting the 591
unpredictability or seeming randomness of the individual outcome. Primes have been found 592
by many to show „deterministic randomness‟ [66-70].
593
We define „real randomness‟ as an unpredictable outcome of a lawless/arbitrary 594
process like selecting a prime number from an infinity of numbers by playing a dice. Here the 595
dice throw per se is not lawless/arbitrary/random. The lawless/arbitrary/random component in 596
a lawless process involving the dice is connecting the dice arbitrarily with a meaningful 597
concept or event that has no lawful connection to the dice, such as connecting prime numbers 598
with the landing of a dice. For a process that involves both a lawful component (dice throw 599
per se) and a lawless component (arbitrarily linking landing of dice with calling a number 600
prime), the process is effectively lawless/arbitrary/random.
601
Both seeming-randomness and real-randomness are unpredictable but the error 602
margins from a pattern are greater with real-randomness. A population of lawfully caused and 603
predictable outcomes follows a precise pattern without any error margin. A population of 604
lawfully caused but unpredictable outcomes follows a less precise pattern with some error 605
margin like the square root of N. A population of lawlessly caused outcomes follows a rough 606
pattern with huge error margins which could be so high as to render the pattern meaningless 607
or equivalent to no pattern. If whatever number that is selected from an infinity of numbers by 608
playing a dice is defined as primes, we would obviously detect no meaningful patterns of 609
primes in most cases, which is equivalent to saying that we could only have patterns with 610
huge error margins. The error margin for a pattern of outcomes that are lawfully caused but 611
unpredictable must necessarily be the smallest among patterns that cannot predict individual 612
outcomes. Any smaller error margin would mean some degree of predictability. If we know 613
that certain position of the tossing hand could cause a higher chance of landing heads while 614
another position favoring tails, we could improve on the error margin but then the coin toss 615
would not qualify as truly unpredictable. True unpredictability is shared by all kinds of 616
randomness. Among these, the seeming-randomness or unpredictability of outcomes of a 617
fully lawful process has the least amount of randomness or the smallest error margin from a 618
regularity pattern.
619
There is a pattern that a fair coin toss follows, which says that the number of heads is 620
equal to half of the number of toss N with an error of the square root of N. This pattern is a 621
law that is valid based on logical reasoning alone. A fair coin toss must not have irregular or 622
arbitrary/random bias toward the head or tail. Each landing of head or tail is fully determined 623
by laws, such as the gravitational law, the exact position of the tossing hand, the wind, etc. A 624
lawful process should produce reproducible outcomes. A coin toss is reproducible if the 625
tossing conditions can be exactly reproduced. A coin toss is only seemingly random because 626
of unpredictability. It is unpredictable because humans cannot measure all the physical 627
parameters that determine the fall of a coin. Also, the laws are not biased to favor of either 628
head or tail and remain unchanged timelessly. If a divine were to suddenly intervene for no 629
reason to cause more landing of the head, the coin toss would be lawlessly caused and would 630
display much wider error margin. If we only detect a seeming randomness in our coin toss 631
with an error of the square root of N, we would be confident that everything is well and 632
regular and no laws have been broken by either random accidents or deliberate intentions. But 633
if we see a much wider variation than the square root of N, we would know that something is 634
wrong or that some laws have been broken either accidentally or deliberately. The coin toss 635
would be considered as unfair.
636
Since the RH means that the error margin from the pattern Li(N) is similar to the coin 637
toss, one can prove the RH by showing that primes must have a regularity pattern which must 638
have an error margin similar to the coin toss. This would be case if one can show that primes 639
are lawfully caused and yet unpredictable. The creation process of prime is lawful and 640
non-random or non-arbitrary. But the outcome of this process, i.e., calling a number a prime, 641
is unpredictable. Because of the unpredictability, one simply cannot have a pattern of primes 642
that is free of error margins. A pattern without error margins would mean predictability.
643
However, because of the lawfulness of the process, the error margins must be the smallest 644
possible among all kinds of unpredictable outcomes, which include those caused by either 645
lawfully or lawlessly determined processes. Specifically, it must be smaller than the error 646
margin of outcomes that involve a lawless process such as arbitrarily calling 6 a prime or 647
calling the head of coin prime. The lawful creation of primes is similar to the coin toss in 648
terms of being lawful and yet unpredictable. It is therefore expected that the two phenomena 649
should have similar error margins. In both cases, the error margins are the smallest possible.
650
Any bigger error margin would mean some degree of lawlessness in the process of creating 651
primes or in the process of coin toss.
652
All other methods of generating or finding primes, such as division by smaller 653
numbers and the sieve of Eratosthenes, are also orderly or deterministic. But unlike the Prime 654
Law here, these methods cannot prove the unpredictability of primes. They define primes by 655
the process of generating primes and thus do not give primes any meaning that is independent 656
of the process: primes are whatever that are found by the process. Under the Prime Law, 657
however, primes have the meaning of uniqueness/uniformity that is independent of the 658
process of creating primes by the Prime Law. Uniqueness has meanings that are independent 659
of the process of creating uniqueness. A creature has meanings that are independent of the 660
process of creating the creature. Unpredictability is a property of the outcome and is 661
independent of the process leading to the outcome. Both lawful and lawless processes can 662
lead to unpredictable outcomes. That a coin falls half of the time head is an inherent property 663
of the coin and is independent of the process of coin toss. If the property of the outcome is all 664
defined or given by the process, then a lawful process simply cannot give the outcome the 665
property of unpredictability or seeming randomness.
666
We have shown that primes must be unique and hence unpredictable because we can 667
use the Prime Law to create primes by merely creating what is unique or unpredictable.
668 669
Evolution and the Prime Law 670
To understand evolution in nature, it may be a productive approach to first 671
understand creative evolution in humans. One can then test if such an understanding may 672
equally explain evolution in nature. If not, one at least would have succeeded in narrowing 673
the field of possible solutions by excluding a major possibility. If intention can be excluded, 674
then lawless chance creation theory, either the intention-less Darwinian theory or the arbitrary 675
God theory of religions, may become valid by default as it would be the only alternative 676
besides a law involving intentions. Remarkably, all known observations indicate that the 677
Prime Law here derived by studying human creativity may well explain evolution in nature.
678
There is a remarkable unity between the MGD theory and the Prime Law. Both involve linear 679
and nonlinear processes. Both require maximum saturation to be reached in the linear process 680
before an event in the nonlinear process can take place, which therefore can account for the 681
nearly constant and yet discontinuous creations of novelty in relation to time. The first 682
individual of a newly evolved novel species as a result of the nonlinear macro-evolutionary 683
process could be viewed as a prime number, while the descendants or followers of the first 684
individual as formed by the linear micro-evolutionary process could be viewed as composite 685
numbers.
686
To further establish the role of the Prime Law in evolution, one could aim to 687
demonstrate or strengthen the following. 1. All fundamentally novel species as a result of the 688
nonlinear macro-evolutionary process were unique at the time when the first individuals of 689
the species first appeared. There were no repeated creations of the same kind of species as 690
such repeats would be copies rather than unique. 2. The wide and persistent existence of 691
certain abstract and gratuitous patterns or beauties in nature that have no apparent functional 692
relevance, such as the Golden Ratio or 5 toes rather than 4 or 6 (pentadactyl pattern), may be 693
because they are the most unique. The Prime Law offers a viable angle to understand this 694
mystery while the chance theories of Darwin and Kimura are completely clueless [71]. 3.
695
There may be only one unique universe and the fine tuning or just right property of our 696
universe may be a result of uniformity selection. Just right is unique. It has long been noted 697
by physicists that the values of over a dozen fundamental physical constants of the universe 698
are precisely fined tuned for life to exist [72]. If the values are slightly different, life could not 699
exist [73]. 4. Most things in nature are at stable maximum or optimum (Pareto optimum) 700
saturation balance. 5. The Principle of Least Action is a most fundamental and unifying 701
physical law of nature and may be embedded in the notion of the Prime Law that only the 702
unique (and its followers) exists in nature. The minimum is unique. 6. Nature is 703
comprehensible, as Einstein put it famously: “The most incomprehensible thing about the 704
universe is that it is comprehensible.” This easily follows if the same law underlies both 705
creative evolutions in nature and human creativity. 7. Nature is written in the language of 706
mathematics. It thus follows that the foundation of mathematics, the primes and the Prime 707
Law, should also be the foundation for the universe. 8. Only intention rather than chance can 708
cause significant reduction in randomness (in the genomes) as found during complexity 709
increases in macro-evolution (Fig. 1A). Intention or mind is inherent in the Prime Law. A 710
chance creation theory or an omnipotent God theory capable of any arbitrary or unlawful 711
actions as described by the major religions would mean the exact opposites of all of the 712
above.
713
As primes are infinite, the Prime Law means that the creation process would be 714
endless. If human creativity and evolution in nature use different laws, one intentional and the 715
other chance or arbitrary, then human creative evolutionary process could be prematurely 716
stopped by chance. Thus, for human creative process to be endless and thus meaningful, it is 717
essential that the same Prime Law is also the foundation for evolution in nature.
718 719
Summary:
720
Evolution, human creations, and prime numbers share a common feature of being 721
both seemingly random (unpredictable) and orderly. They all also involve both linear and 722
nonlinear processes, and show a similar pattern of nearly constant and yet seemingly random 723
creation of novelty in relation to time. Such characteristics are inconsistent with chance or 724
arbitrary creations and can be explained by a creation algorithm that nonetheless cannot be 725
predictive. This algorithm appears to be hardwired in the human brain.
726
727
Acknowledgements:
728
Supported by the National Natural Science Foundation of China grant 81171880 and the 729
National Basic Research Program of China grant 2011CB51001.
730 731
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